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arXiv · 2403.13997

A Geometric flow towards hamiltonian stationary submanifolds

Jingyi Chen·Micah Warren
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Abstract

In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary Lagrangian submanifolds. The flow is not strictly parabolic but it corresponds to a fourth order strictly parabolic scalar equation in the cotangent bundle of the submanifold via Weinstein's Lagrangian neighborhood theorem. For any compact initial Lagrangian immersion, we establish short-time existence, uniqueness, and higher order estimates when the second fundamental forms are uniformly bounded up to time $T$.

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Jingyi Chen, Micah Warren. 2024-03-20. A Geometric flow towards hamiltonian stationary submanifolds. https://arxiv.org/abs/2403.13997

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